Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories

In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massive O($N$) $\lambda\phi^{4}$ scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and latter at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For attaining that goal, we employ three different and independent field-theoretic renormalization group methods. The results found for the critical exponents show that they are identical in the three distinct methods and equal to their Lorentz invariant counterparts. Furthermore, we show that the results obtained here, based on the single concept of loop order of the referred terms of the corresponding $\beta$-function and anomalous dimensions, reduce to the ones obtained through the earlier non-exact approach based on a joint redefinition of the field and coupling constant of the theory, in the appropriate limit.